We consider Markov processes $z_t=\{x_t,y_t\}$ in a product space $X\times Y$ ( $x_t\in X$ , $y_t\in Y$ ), $Y$ being a finite-dimensional Euclidean space. Such a process is called a process with homogeneous second component if its transition probability function $P(t,x,y,s,A,B)$ , $x\in X$ , $y\in Y$ , $A\subset X$ , $B\subset Y$ , $t<s$ , satisfies the condition $$ P(t,x,y,s,A,B)=P(t,x,0,s,A,B_{-y}), $$where$B_{-y}$is the set of$y'$'s such that$y+y'\in B$. In §1 we study general properties of such processes. In §2 the case is considered when$x_t$is a process with denumerable set of states. §3 deals with time-homogeneous processes.